Found problems: 85335
2011 IMO Shortlist, 3
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
2005 Alexandru Myller, 2
Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$.
[i]Mihai Piticari[/i]
1999 All-Russian Olympiad, 6
Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.
TNO 2008 Junior, 7
A $5 \times 5$ grid is given, called $f_1$:
\[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
-1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
\end{array}
\]
A new grid $f_{n+1}$ is constructed where each cell is equal to the product of its neighboring cells in grid $f_n$.
(a) Find the grids $f_6$ and $f_7$.
(b) Find the grids $f_{2008}$ and $f_{2009}$.
(c) Find $f_{2n}$ and $f_{2n+1}$ for any $n \in \mathbb{N}$.
*Note: Neighboring cells are those that share an edge, not just a vertex.*
2024 Middle European Mathematical Olympiad, 4
Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)
1971 Spain Mathematical Olympiad, 3
If $0 < p$, $0 < q$ and $p +q < 1$ prove $$(px + qy)^2 \le px^2 + qy^2$$
Kvant 2021, M2640
In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.
PEN S Problems, 34
Let $S_{n}$ be the sum of the digits of $2^n$. Prove or disprove that $S_{n+1}=S_{n}$ for some positive integer $n$.
PEN H Problems, 61
Solve the equation $2^x -5 =11^{y}$ in positive integers.
2022 Moldova Team Selection Test, 6
Let $A$ be a point outside of the circle $\Omega$. Tangents from $A$ touch $\Omega$ in points $B$ and $C$. Point $C$, collinear with $A$ and $P$, is between $A$ and $P$, such that the circumcircle of triangle $ABP$ intersects $\Omega$ again in point $E$. Point $Q$ is on the segment $BP$, such that $\angle PEQ=2 \cdot \angle APB$. Prove that the lines $BP$ and $CQ$ are perpendicular.
2011 All-Russian Olympiad, 1
In every cell of a table with $n$ rows and ten columns, a digit is written. It is known that for every row $A$ and any two columns, you can always find a row that has different digits from $A$ only when it intersects with two columns. Prove that $n\geq512$.
2022 Serbia Team Selection Test, P3
Let $n$ be an odd positive integer. Given are $n$ balls - black and white, placed on a circle. For a integer $1\leq k \leq n-1$, call $f(k)$ the number of balls, such that after shifting them with $k$ positions clockwise, their color doesn't change.
a) Prove that for all $n$, there is a $k$ with $f(k) \geq \frac{n-1}{2}$.
b) Prove that there are infinitely many $n$ (and corresponding colorings for them) such that $f(k)\leq \frac{n-1}{2}$ for all $k$.
2015 Junior Balkan Team Selection Test, 4
The diagonals $AD$, $BE$, $CF$ of cyclic hexagon $ABCDEF$ intersect in $S$ and $AB$ is parallel to $CF$ and lines $DE$ and $CF$ intersect each other in $M$. Let $N$ be a point such that $M$ is the midpoint of $SN$. Prove that circumcircle of $\triangle ADN$ is passing through midpoint of segment $CF$.
2005 National Olympiad First Round, 13
Let $ABCD$ be an isosceles trapezoid such that its diagonal is $\sqrt 3$ and its base angle is $60^\circ$, where $AD \parallel BC$. Let $P$ be a point on the plane of the trapezoid such that $|PA|=1$ and $|PD|=3$. Which of the following can be the length of $[PC]$?
$
\textbf{(A)}\ \sqrt 6
\qquad\textbf{(B)}\ 2\sqrt 2
\qquad\textbf{(C)}\ 2 \sqrt 3
\qquad\textbf{(D)}\ 3\sqrt 3
\qquad\textbf{(E)}\ \sqrt 7
$
1981 Yugoslav Team Selection Test, Problem 3
Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that
\begin{align*}
x+2y+3z+7t&=a,\\
y+2z+5t&=b.
\end{align*}
2014 ELMO Shortlist, 13
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
1964 Dutch Mathematical Olympiad, 5
Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$:
(a) $g_{n+1}\le g_n$
(b) $g_{10}= 000$.
LMT Speed Rounds, 2011.15
Given that $20N^2$ is a divisor of $11!,$ what is the greatest possible integer value of $N?$
2018 India Regional Mathematical Olympiad, 6
Define a sequence $\{a_n\}_{n\geq 1}$ of real numbers by \[a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.\] Prove that \[\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1\] for every natural number $N$.
2020 AMC 10, 3
Assuming $a\neq3$, $b\neq4$, and $c\neq5$, what is the value in simplest form of the following expression?
$$\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$$
$\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}$
2014 Contests, 2
We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex.
Find all triangulable numbers.
1993 Tournament Of Towns, (370) 2
Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the intersection point of the lines $AB$ and $CD$ and $N$ is the intersection point of the lines $BC$ and $AD$. It is known that $BM = DN$. Prove that $CM = CN$.
(F Nazarov)
2006 Harvard-MIT Mathematics Tournament, 3
The train schedule in Hummut is hopelessly unreliable. Train $A$ will enter Intersection $X$ from the west at a random time between $9:00$ am and $2:30$ pm; each moment in that interval is equally likely. Train $B$ will enter the same intersection from the north at a random time between $9:30$ am and $12:30$ pm, independent of Train $A$; again, each moment in the interval is equally likely. If each train takes $45$ minutes to clear the intersection, what is the probability of a collision today?
1989 China National Olympiad, 4
Given a triangle $ABC$, points $D,E,F$ lie on sides $BC,CA,AB$ respectively. Moreover, the radii of incircles of $\triangle AEF, \triangle BFD, \triangle CDE$ are equal to $r$. Denote by $r_0$ and $R$ the radii of incircles of $\triangle DEF$ and $\triangle ABC$ respectively. Prove that $r+r_0=R$.
2009 Iran MO (3rd Round), 4
Does there exists two functions $f,g :\mathbb{R}\rightarrow \mathbb{R}$ such that:
$\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1$
Time allowed for this problem was 75 minutes.